Some related problems:
My problem comes from one step of a certain proof:
$\|Av\|^2=(Av)^T(Av)=v^TA^TAv=v^TIv = v^Tv =\|v\|^2 \Rightarrow \|Av\| = \|v\|$
Note: $A$ is an orthogonal matrix.
More general one:
$\|Av\|^2=\langle Av,Av \rangle=\langle v,A^TAv \rangle=\langle v,Iv \rangle= \langle v,v\rangle=\|v\|^2 \Rightarrow \|Av\| = \|v\|$
They look similar.
It seems the above equality only holds on $l_2$ norm(the first case) and inner product induced norm (the second case).
So such equality cannot hold for general norm, right?
I just want to make my proof more strict.